Skip to main content
Springer Nature Link
Log in

The ASEP and Determinantal Point Processes

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show how such ensembles arise as limits of various hypergeometric orthogonal polynomial ensembles. We then prove that the q-Laplace transform of the height function of the ASEP with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete Laguerre ensemble—a member of the new family. This allows us to obtain the large time asymptotics of the ASEP in three limit regimes: (a) for finitely many rightmost particles; (b) GUE Tracy–Widom asymptotics of the height function; (c) KPZ asymptotics of the height function for the ASEP with weak asymmetry. We also give similar results for two instances of the stochastic six vertex model in a quadrant. The proofs are based on limit transitions for the corresponding determinantal point processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aggarwal, A.: Convergence of the Stochastic Six-Vertex Model to the ASEP. Math. Phys. Anal. Geom. (2017) 20(3). doi:10.1007/s11040-016-9235-8

  2. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Fizmat, Moscow, 1961. English translation: Oliver and Boyd Ltd, Edinburgh and London (1965)

  3. Amir G., Corwin I., Quastel J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Commun. Pure Appl. Math. 64, 466–537 (2011) arXiv:1003.0443

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertini L., Giacomin G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bornemann F: On the scaling limits of determinantal point processes with kernels induced by Sturm–Liouville operators. SIGMA 12, 083 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Borodin A.: Duality of orthogonal polynomials on a finite set. J. Stat. Phys. 109(5), 1109–1120 (2002) arXiv:math/0101125

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Borodin, A.: Stochastic Higher Spin Six Vertex Model and Macdonald Measures. arXiv:1608.01553

  8. Borodin A., Corwin I.: Macdonald processes. Probab. Theory Relat. Fields 158(1–2), 225–400 (2014) arXiv:1111.4408

    Article  MathSciNet  MATH  Google Scholar 

  9. Borodin A., Corwin I., Gorin V.: Stochastic six-vertex model. Duke Math. J. 165(3), 563–624 (2016) arXiv:1407.6729

    Article  MathSciNet  MATH  Google Scholar 

  10. Borodin A., Corwin I., Sasamoto T.: From duality to determinants for q-TASEP and ASEP. Ann. Prob. 42(6), 2314–2382 (2014) arXiv:1207.5035

    Article  MathSciNet  MATH  Google Scholar 

  11. Borodin A., Gorin V.: Shuffling algorithm for boxed plane partitions. Adv. Math. 220, 1739–1770 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Borodin, A., Gorin, V.: Lectures on integrable probability. In: Probability and Statistical Physics in St. Petersburg. Proceedings of Symposia in Pure Mathematics, vol. 91, pp. 162–221. Amer. Math. Soc. (2016). arXiv:1212.3351

  13. Borodin, A., Gorin, V.: Moments match between the KPZ equation and the Airy point process. SIGMA 12, 102 (2016). arXiv:1608.01557

  14. Borodin A., Gorin V., Rains E.: q-Distributions on boxed plane partitions. Sel. Math. New Ser. 16(4), 731–789 (2010) arXiv:0905.0679

    Article  MathSciNet  MATH  Google Scholar 

  15. Borodin A., Kuan J.: Random surface growth with a wall and Plancherel measures for \({O(\infty)}\). Commun. Pure Appl. Math. 63(7), 831–894 (2010) arXiv:0904.2607

    Article  MathSciNet  MATH  Google Scholar 

  16. Borodin A., Okounkov A., Olshanski G.: Asymptotics of Plancherel measures for symmetric groups. J. Am. Math. Soc. 13(3), 481–515 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Borodin A., Olshanski G.: Asymptotics of Plancherel-type random partitions. J. Algebra 313(1), 40–60 (2007) arXiv:math/0610240

    Article  MathSciNet  MATH  Google Scholar 

  18. Borodin A., Olshanski G.: Meixner polynomials and random partitions. Mosc. Math. J. 6(4), 629–655 (2006) arXiv:math/0609806

    MathSciNet  MATH  Google Scholar 

  19. Borodin, A., Petrov, L.: Higher Spin Six Vertex Model and Symmetric Rational Functions. Sel. Math. New. Ser. (to appear). doi:10.1007/s00029-016-0301-7. arXiv:1605.01349

  20. Borodin, A., Petrov, L.: Lectures on Integrable Probability: Stochastic Vertex Models and Symmetric Functions. arXiv:1605.01349

  21. Breuer, J., Duits, M.: Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients. J. Am. Math. Soc. 30, 27–66 (2017). arXiv:1309.6224

  22. Breuer J., Duits M.: Universality of Mesoscopic fluctuations for orthogonal polynomial ensembles. Commun. Math. Phys. 342(2), 491–531 (2016) arXiv:1411.5205

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Calabrese P., Le Doussal P., Rosso A.: Free-energy distribution of the directed polymer at high temperature. Euro. Phys. Lett. 90, 20002 (2010)

    Article  ADS  Google Scholar 

  24. Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1(1) (2012). arXiv:1106.1596

  25. Corwin I., Petrov L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 343(2), 651–700 (2016) arXiv:1502.07374

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Corwin, I., Tsai, L.-C.: KPZ equation limit of higher-spin exclusion processes. Ann. Probab. to appear. arXiv:1505.04158

  27. Deift, P.: Integrable operators. In: Buslaev, V., Solomyak, M., Yafaev, D. (eds) Differential Operators and Spectral Theory: M. Sh. Birman’s 70th Anniversary Collection. American Mathematical Society Translations, Ser. 2, vol. 189. AMS, Providence, RI (1999)

  28. Dotsenko V.: Bethe ansatz derivation of the Tracy–Widom distribution for one-dimensional directed polymers. Euro. Phys. Lett. 90, 20003 (2010)

    Article  ADS  Google Scholar 

  29. Edelman A., Sutton B.D.: From random matrices to stochastic operators. J. Stat. Phys. 127(6), 1121–1165 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Ferrari, P.L., Spohn, H.: Random growth models. In: Akemann, G., Baik, J., Di Francesco, P. (eds). The Oxford Handbook of Random Matrix Theory. Oxford University Press (2011). arXiv:1003.0881

  31. Gorin V.: Non-intersecting paths and Hahn orthogonal polynomial ensemble. Funct. Anal. Appl. 42(3), 180–197 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Grünbaum, F.A.: The bispectral problem: an overview. In: Special Functions 2000: Current Perspective and Future Directions. NATO Science Series Volume 30, Springer, pp. 129–140 (2001)

  33. Gwa L.-H., Spohn H.: Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys. Rev. Lett. 68(6), 725–728 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Imamura T., Sasamoto T.: Determinantal structures in the O’Connell–Yor directed random polymer model. J. Stat. Phys. 163(4), 675–713 (2016) arXiv:1506.05548

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Its A.R., Izergin A.G., Korepin V.E., Slavnov N.A.: Differential equations for quantum correlation functions. Int. J. Mod. Phys. B 4, 10037–1037 (1990)

  36. Johansson K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) 163(4), 675–713 (2016) arXiv:math/9903134

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Johansson, K.: Random matrices and determinantal processes. Mathematical Statistical Physics, Session LXXXIII: Lecture Notes of the Les Houches Summer School, pp. 1–56 (2005). arXiv:math-ph/0510038

  38. Koekoek, R., Swarttouw, R.F.: The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue, Report no. 98-17 (1998), Delft University of Technology. http://homepage.tudelft.nl/11r49/askey/

  39. Koenig W.: Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2, 385–447 (2005) arXiv:math/0403090

    Article  MathSciNet  Google Scholar 

  40. Liggett T.M.: Interacting Particle Systems. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  41. Macdonald I.G.: Symmetric functions and Hall polynomials, 2nd edn. The Clarendon Press, Oxford University Press, New York (1995)

    Google Scholar 

  42. MacDonald J., Gibbs J., Pipkin A.: Kinetics of biopolymerization on nucleic acid templates. Biopolymers 6, 1–25 (1968)

    Article  Google Scholar 

  43. O’Connell N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40(2), 437–458 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. O’Connell, N., Yor, M.: A Representation for Non-Colliding Random Walks. Electr. Commun. Probab. 7 (2002), paper no. 1, 1–12

  45. Okounkov A.: Infinite wedge and random partitions. Sel. Math. 7, 57–81 (2001) arXiv:math/9907127

    Article  MathSciNet  MATH  Google Scholar 

  46. Olshanski G.: Difference operators and determinantal point processes. Funct. Anal. Appl. 42(4), 317–329 (2008) arXiv:0810.3751

    Article  MathSciNet  MATH  Google Scholar 

  47. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. I. Functional Analysis, Academic Press, New York (1972)

  48. Sasamoto T., Spohn H.: One-dimensional KPZ equation: an exact solution and its universality. Phys. Rev. Lett. 104, 230602 (2010)

    Article  ADS  Google Scholar 

  49. Sasamoto T., Spohn H.: The crossover regime for the weakly asymmetric simple exclusion process. J. Stat. Phys. 140, 209–231 (2010) arXiv:1002.1873

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Simon B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  51. Soshnikov A.: Determinantal random point fields. Russ. Math. Surv. 55(5), 923–975 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  52. Spitzer F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  53. Tao, T.: The Dyson and Airy Kernels of GUE Via Semiclassical Analysis (Tao’s blog, October 23, 2010), https://terrytao.wordpress.com/2010/10/23/the-dyson-and-airy-kernels-of-gue-via-semiclassical-analysis/

  54. Tao, T.: Topics in Random Matrix theory. Graduate Studies in Mathematics, vol. 132. Amer. Math. Soc. (2012)

  55. Tracy C., Widom H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159(1), 151–174 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Tracy C., Widom H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132, 291–300 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Tracy C., Widom H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, MA, USA

    Alexei Borodin

  2. Institute for Information Transmission Problems, Moscow, Russia

    Alexei Borodin & Grigori Olshanski

  3. National Research University Higher School of Economics, Moscow, Russia

    Grigori Olshanski

Authors
  1. Alexei Borodin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Grigori Olshanski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexei Borodin.

Additional information

Communicated by P. Deift

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borodin, A., Olshanski, G. The ASEP and Determinantal Point Processes. Commun. Math. Phys. 353, 853–903 (2017). https://doi.org/10.1007/s00220-017-2858-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2858-1

Search

Navigation